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DISCOVERY OF GREAT PYRAMID MATH
L. Kaliambos (Kaliambos-Natural Philosophy) 3 December 2018 Today writing in Google the topic “Discovery of Great Pyramid math” we see that a dominant article entitled “ Pi and the great pyramid ” starts with the discovery of John Taylor (1859), who first proposed the idea that the number &pi might have been intentionally incorporated into the design of the Great Pyramid of Khufu at Giza. However the article is full of many assumptions which lead to complications. In this research I have attempted to incorporate some of the conclusions drawn from an illuminating experience about my combinatory method used for revealing the math of Parthenon, of Caryatids, of the walls of Alexandria in Egypt, and of the Hephaestion tomb in Amphipolis. (MATHEMATICAL TOMB OF HEPHAESTION). Since the Great Pyramid has the shape of a tetragonal pyramid purposely I present here the figure of a square pyramid taken from Wikipedia with the base length, α, the lateral side, e, the slant height, S, and the height, h. Then, for a careful comparison using this figure with the real dimensions of Great Pyramid we see that the most useful is the article "Great Pyramid of Giza -Wikipedia", because it describes in details the side of the square α = 230.34 m and the initial height h = 146.5 m. However in the absence of a knowledge about the golden numbers Φ = 1.618.. and Φ0.5 = 1.272… used by the Egyptian mathematicians, Wikipedia cannot provide any information about the ratio h/α which reveals the golden number Φ0.5 = 1.272.. That is h/α = 146.5/230.34 = 0.636.. = (1.272..) /2 = Φ0.5 /2 It means that the Egyptian mathematicians in order to design the Great Pyramid (2560 BC) on the ground used a theoretical cone pyramid with a circle c of a radius r = α/2 = 115.17 m inscribed in the square of the base. Therefore to design the Great Pyramid on the ground they used the ratio h/r = 146.15/115.17 = 1.272 = Φ0.5 . Then under detailed measurements on the ground they should find that the circumference c inscribed in the square is equal to 6.28... units of length. That is, they were able to find the ratio c/r = 2π = (6.28…). Ιn my article “Discovery of Φ and π in Giza great pyramid” using a combinatory method I showed that both Pi = π = c/d = 3.14…and the Φ = (1 +50.5)/2 =1.618… were used in the design of the pyramid because it was thought that the golden numbers Φ and Φ0.5 and also the ratio c/r = 2π = ( 6.28 ..) are sacred numbers in a relationship. For example the Egyptian mathematicians of the Great pyramid found that π is given by π = 4/Φ0.5 .= 3.1446… which is close to the modern value of π = 3.14159… Also they found that the diagonal Φ = 1.618.. of a pentagonal scheme measured in units of length is given by Φ = Φ2 - 1 or 1.618 = (1.618)(1.618) - 1 = 2.618 - 1 However, one should ask how the ancient Egyptians should be able to find the solution of the quadratic formula Φ2 - Φ -1 = 0 to get accurately that Φ = (1+ 50.5)/2 = 1.61803398874.. The study of this called algebra goes to the antiquity. Recent discoveries have shown that Babylonians and Egyptians solved problems in algebra , although they had no symbols for variables. They used only words to indicate such numbers, and for that reason their algebra has been referred to as theoretical algebra. The Ahmes Papyrus, an Egyptian scroll going back to 1600 BC has a number of problems in algebra, in which the unknown is referred to as a hau, meaning “a heap”. Also practically the so -called Pythagorean theorem (Greek mathematics of 6thcentury BC) was well known to Babylonians and Egyptians. Thus writing 1 + Φ = Φ2 we may write also (1)2 + (Φ0.5)2 = Φ2 In this case one sees that for a sacred cone of radius r, height h = rΦ0.5 and slant height S = rΦ using the Pythagorean theorem we may write r2 + h2 = S2 or r2 + (hΦ0.5)2 = (rΦ)2 Then for r = 1 (unit of length) one gets 1 + Φ = Φ2 In such a cone when r = 1 (unit of length) under detailed measurements on the ground for designing the square and the inscribed circle of the Great pyramid they should measure the circumference c = 2π = (6.28..). In other words such a cone pyramid was believed to be a sacred pyramid because it includes the mystic numbers Phi = Φ and Pi = π So in the great pyramid using the values of α = 230.34 m, of r = α/2 = 115.17 m, and of h = 146.5 m, one can calculate the slant height by applying the Pythagorean theorem as S = (r2 + h2)0.5 = (13264,1289 + 21462.25)0.5 = 186.35 m = rΦ = 115.17(1.618…) = 186.35 m. Here one sees that in the absence of a knowledge about the math of the golden ratio of the Egyptian mathematicians in this very simple calculation no one would be able to reveal that the slant height S of the Great Pyramid should contain the golden number Φ. In other words the slant height measured in units of length is equal to Φ. In the same way to calculate the lateral side, e, applying the Pythagorean theorem one gets e = (S2 + α2/4)0.5 = 219.06 m. However the Egyptian mathematicians using S = rΦ for calculating the e they should write e = (r2Φ2 + r2)0.5 = r( Φ2 + 1)0.5 = 219.06 m That is, the lateral side, e, measured in units of length is given by e = (Φ2 +1)0.5 It is of intererest to notice that the same result measured in units of length by the Egyptian mathematicians could be calculated by using the height h = rΦ0.5 and the half of the diagonal δ of the square. That is δ/2 = 20.5α/2 = 20.5r In this case since r =1 applying the Pythagorean theorem they should get e2 = (Φ0.5)2 + (20.5)2 Or e = (Φ +2)0.5 = (Φ +1 +1)0.5 Since Φ +1 = Φ2 they got e = (Φ2 +1)0.5 Then comparing the perimeter of the square P = 4α = 8r with the r they should get P/r = 8, while comparing the P with the half diagonal (δ/2) they should get P/20.5r = 8/(1.4142..) = 5.6569.. In this case a great circle, in which the square is inscribed inside) has a circumference equal to (6.28..)20.5 = 8.88.. units of length. However for a circle C with a radius R > r but smaller than 20.5r , in which the circumference C is equal to P = 8 units of length, one observes that the ratio C/R = 6.28.. is greater than P/20.5r and smaller than P/r. In this case the Egyptian mathematicians surprisingly found with detailed measurements on the ground that such a circle of C = P = 8 units of length could be drawn when the radius R is equal to Φ0.5 . That is C/R = P/Φ0.5 = 8/1.272.. = 6.289.. or π = C/d = 4/Φ0.5 = 3.1446… which is close to π = 3.14159… For such a radius R = rΦ0.5 = h in the article "Mathematical Encoding in the Great Pyramid " we read: "A circle drawn from the center with radius equal to the height of the pyramid has a circumference equal to the square’s perimeter. Thus, the Great Pyramid squares the circle. Its form approximates the solution sought by ancient geometers, who for many generations endeavored to square the circle by length (squaring a circle by area is a related problem). Note: exact squaring of the circle is impossible due to the transcendental qualities of π (proved by Lindemann in 1882)." In fact, for C = P = 8 units of length today applying the modern value of π = (3.14159…) we get R = P/2π = 8/2(3.14159..) = 8/(6.28318) = (1.27324..), while Φ0.5 = (1.272…) In other words the Egyptian mathematicians were absolutely correct for designing in units of length the α/2 = r = 1, the height, h = Φ0.5 = (1.272..) the slant height, S = Φ = (1.618) and the lateral side, e = (Φ2 + 1)0.5 (1.902..). On the other hand for calculating the radius R of the circle for C = P = 8 units of length the error of the radius R was very small, because (1.27324..) - (1.272..) = (0.00124..). In the same way for calculating the mystic number π the small error was (3.14465..) - (3.14159..) = (0.00306..). It is of interesting to note that for calculating the volume of pyramid the Egyptian architects should write V = (1/3)α2h = (1/3)(4r2)(rΦ0.5) = (4/3)r3Φ0.5 In this formula writing r =1 (unit of length) one gets the very simple formula V= (4/3)Φ0.5 which includes not only the mystic number Φ0.5 but also the mystic numbers 3 and 4 of the so-called Pythagorean theorem of the Greek mathematics. Surprisingly today using geometry we see that the volume V = (1/3) (πr2h) of the secret cone of Egyptians with radius r and height h = rΦ0.5 should be given by V = (1/3)(πr2h) = (1/3)(4/Φ0.5 ) r2(rΦ0.5) = (4/3)r3 Then for r =1 we get V = 4/3 In other words the volume V= 4/3 of the Egyptian secret cone contains only the secret numbers 4 and 3 of the Pythagorean theorem of the Greek mathematics. To conclude I emphasize that the Egyptian mathematicians using a theoretical cone with radius r =1, and giving a height h = Φ0.5 and a slant height S = Φ, they succeeded to build in 2560 BC the MATHEMATICAL GREAT PYRAMID with the mystic numbers Φ, Φ0.5 and π including also not only the so-called Pythagorean theorem but also the numbers 3 and 4 of the same theorem. Such mystic numbers used later (438 BC) by Phidias for building the MATHEMATICAL PARTHENON. (Correct history of math).